mmtheorems8 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12229

Statement List for Metamath Proof Explorer - 701-800 - Page 8 of 123

Type	Label	Description
Statement
Theorem	mp2an 701	An inference based on modus ponens.
|- ph   &   |- ps   &   |- ((ph /\ ps) -> ch)   =>   |- ch
Theorem	mpani 702	An inference based on modus ponens.
|- ps   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ch -> th))
Theorem	mpan2i 703	An inference based on modus ponens.
|- ch   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ps -> th))
Theorem	mp2ani 704	An inference based on modus ponens.
|- ps   &   |- ch   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> th)
Theorem	mpand 705	A deduction based on modus ponens.
|- (ph -> ps)   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ch -> th))
Theorem	mpan2d 706	A deduction based on modus ponens.
|- (ph -> ch)   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ps -> th))
Theorem	mp2and 707	A deduction based on modus ponens.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> th)
Theorem	mpdan 708	An inference based on modus ponens.
|- (ph -> ps)   &   |- ((ph /\ ps) -> ch)   =>   |- (ph -> ch)
Theorem	mpancom 709	An inference based on modus ponens with commutation of antecedents.
|- (ps -> ph)   &   |- ((ph /\ ps) -> ch)   =>   |- (ps -> ch)
Theorem	mpanl1 710	An inference based on modus ponens.
|- ph   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ps /\ ch) -> th)
Theorem	mpanl2 711	An inference based on modus ponens.
|- ps   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mpanl12 712	An inference based on modus ponens.
|- ph   &   |- ps   &   |- (((ph /\ ps) /\ ch) -> th)   =>   |- (ch -> th)
Theorem	mpanr1 713	An inference based on modus ponens.
|- ps   &   |- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ch) -> th)
Theorem	mpanr2 714	An inference based on modus ponens.
|- ch   &   |- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ps) -> th)
Theorem	mpanr12 715	An inference based on modus ponens.
|- ps   &   |- ch   &   |- ((ph /\ (ps /\ ch)) -> th)   =>   |- (ph -> th)
Theorem	mpanlr1 716	An inference based on modus ponens.
|- ps   &   |- (((ph /\ (ps /\ ch)) /\ th) -> ta)   =>   |- (((ph /\ ch) /\ th) -> ta)
Theorem	mtt 717	Modus-tollens-like theorem.
|- (-. ph -> (-. ps <-> (ps -> ph)))
Theorem	mt2bi 718	A false consequent falsifies an antecedent.
|- ph   =>   |- (-. ps <-> (ps -> -. ph))
Theorem	mtbid 719	A deduction from a biconditional, similar to modus tollens.
|- (ph -> -. ps)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ch)
Theorem	mtbird 720	A deduction from a biconditional, similar to modus tollens.
|- (ph -> -. ch)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ps)
Theorem	mtbii 721	An inference from a biconditional, similar to modus tollens.
|- -. ps   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ch)
Theorem	mtbiri 722	An inference from a biconditional, similar to modus tollens.
|- -. ch   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> -. ps)
Theorem	2th 723	Two truths are equivalent.
|- ph   &   |- ps   =>   |- (ph <-> ps)
Theorem	2false 724	Two falsehoods are equivalent.
|- -. ph   &   |- -. ps   =>   |- (ph <-> ps)
Theorem	tbt 725	A wff is equivalent to its equivalence with truth. (The proof was shortened by Juha Arpiainen, 19-Jan-2006.)
|- ph   =>   |- (ps <-> (ps <-> ph))
Theorem	nbn2 726	The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.)
|- (-. ph -> (-. ps <-> (ph <-> ps)))
Theorem	nbn 727	The negation of a wff is equivalent to the wff's equivalence to falsehood.
|- -. ph   =>   |- (-. ps <-> (ps <-> ph))
Theorem	nbn3 728	Transfer falsehood via equivalence.
|- ph   =>   |- (-. ps <-> (ps <-> -. ph))
Theorem	biantru 729	A wff is equivalent to its conjunction with truth.
|- ph   =>   |- (ps <-> (ps /\ ph))
Theorem	biantrur 730	A wff is equivalent to its conjunction with truth.
|- ph   =>   |- (ps <-> (ph /\ ps))
Theorem	biantrud 731	A wff is equivalent to its conjunction with truth.
|- (ph -> ps)   =>   |- (ph -> (ch <-> (ch /\ ps)))
Theorem	biantrurd 732	A wff is equivalent to its conjunction with truth.
|- (ph -> ps)   =>   |- (ph -> (ch <-> (ps /\ ch)))
Theorem	mpbiran 733	Detach truth from conjunction in biconditional.
|- (ph <-> (ps /\ ch))   &   |- ps   =>   |- (ph <-> ch)
Theorem	mpbiran2 734	Detach truth from conjunction in biconditional.
|- (ph <-> (ps /\ ch))   &   |- ch   =>   |- (ph <-> ps)
Theorem	mpbir2an 735	Detach a conjunction of truths in a biconditional.
|- (ph <-> (ps /\ ch))   &   |- ps   &   |- ch   =>   |- ph
Theorem	biimt 736	A wff is equivalent to itself with true antecedent.
|- (ph -> (ps <-> (ph -> ps)))
Theorem	pm5.5 737	Theorem *5.5 of [WhiteheadRussell] p. 125.
|- (ph -> ((ph -> ps) <-> ps))
Theorem	pm5.62 738	Theorem *5.62 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 21-Jun-2005.)
|- (((ph /\ ps) \/ -. ps) <-> (ph \/ -. ps))
Theorem	biort 739	A wff is disjoined with truth is true.
|- (ph -> (ph <-> (ph \/ ps)))
Theorem	biorf 740	A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121.
|- (-. ph -> (ps <-> (ph \/ ps)))
Theorem	biorfi 741	A wff is equivalent to its disjunction with falsehood.
|- -. ph   =>   |- (ps <-> (ps \/ ph))
Theorem	bianfi 742	A wff conjoined with falsehood is false.
|- -. ph   =>   |- (ph <-> (ps /\ ph))
Theorem	bianfd 743	A wff conjoined with falsehood is false.
|- (ph -> -. ps)   =>   |- (ph -> (ps <-> (ps /\ ch)))
Theorem	pm4.82 744	Theorem *4.82 of [WhiteheadRussell] p. 122.
|- (((ph -> ps) /\ (ph -> -. ps)) <-> -. ph)
Theorem	pm4.83 745	Theorem *4.83 of [WhiteheadRussell] p. 122.
|- (((ph -> ps) /\ (-. ph -> ps)) <-> ps)
Theorem	pclem6 746	Negation inferred from embedded conjunct.
|- ((ph <-> (ps /\ -. ph)) -> -. ps)
Theorem	biantr 747	A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117.
|- (((ph <-> ps) /\ (ch <-> ps)) -> (ph <-> ch))
Theorem	orbidi 748	Disjunction distributes over the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseer.ist.psu.edu/lifschitz98calculational.html.
|- ((ph \/ (ps <-> ch)) <-> ((ph \/ ps) <-> (ph \/ ch)))
Theorem	biass 749	Associative law for the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseer.ist.psu.edu/lifschitz98calculational.html. Interestingly, this law was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (The proof was shortened by Juha Arpiainen, 19-Jan-2006.)
|- (((ph <-> ps) <-> ch) <-> (ph <-> (ps <-> ch)))
Theorem	biluk 750	Lukasiewicz's shortest axiom for equivalential calculus. Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96.
|- ((ph <-> ps) <-> ((ch <-> ps) <-> (ph <-> ch)))
Theorem	pm5.7 751	Disjunction distributes over the biconditional. Theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbidi 748. (Contributed by Roy F. Longton, 21-Jun-2005.)
|- (((ph \/ ch) <-> (ps \/ ch)) <-> (ch \/ (ph <-> ps)))
Theorem	bigolden 752	Dijkstra-Scholten's Golden Rule for calculational proofs.
|- (((ph /\ ps) <-> ph) <-> (ps <-> (ph \/ ps)))
Theorem	pm5.71 753	Theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.)
|- ((ps -> -. ch) -> (((ph \/ ps) /\ ch) <-> (ph /\ ch)))
Theorem	pm5.75 754	Theorem *5.75 of [WhiteheadRussell] p. 126.
|- (((ch -> -. ps) /\ (ph <-> (ps \/ ch))) -> ((ph /\ -. ps) <-> ch))
Theorem	bimsc1 755	Removal of conjunct from one side of an equivalence.
|- (((ph -> ps) /\ (ch <-> (ps /\ ph))) -> (ch <-> ph))
Theorem	ecase2d 756	Deduction for elimination by cases.
|- (ph -> ps)   &   |- (ph -> -. (ps /\ ch))   &   |- (ph -> -. (ps /\ th))   &   |- (ph -> (ta \/ (ch \/ th)))   =>   |- (ph -> ta)
Theorem	ecase3 757	Inference for elimination by cases.
|- (ph -> ch)   &   |- (ps -> ch)   &   |- (-. (ph \/ ps) -> ch)   =>   |- ch
Theorem	ecase 758	Inference for elimination by cases.
|- (-. ph -> ch)   &   |- (-. ps -> ch)   &   |- ((ph /\ ps) -> ch)   =>   |- ch
Theorem	ecase3d 759	Deduction for elimination by cases.
|- (ph -> (ps -> th))   &   |- (ph -> (ch -> th))   &   |- (ph -> (-. (ps \/ ch) -> th))   =>   |- (ph -> th)
Theorem	ccase 760	Inference for combining cases.
|- ((ph /\ ps) -> ta)   &   |- ((ch /\ ps) -> ta)   &   |- ((ph /\ th) -> ta)   &   |- ((ch /\ th) -> ta)   =>   |- (((ph \/ ch) /\ (ps \/ th)) -> ta)
Theorem	ccased 761	Deduction for combining cases.
|- (ph -> ((ps /\ ch) -> et))   &   |- (ph -> ((th /\ ch) -> et))   &   |- (ph -> ((ps /\ ta) -> et))   &   |- (ph -> ((th /\ ta) -> et))   =>   |- (ph -> (((ps \/ th) /\ (ch \/ ta)) -> et))
Theorem	ccase2 762	Inference for combining cases.
|- ((ph /\ ps) -> ta)   &   |- (ch -> ta)   &   |- (th -> ta)   =>   |- (((ph \/ ch) /\ (ps \/ th)) -> ta)
Theorem	4cases 763	Inference eliminating two antecedents from the four possible cases that result from their true/false combinations.
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ -. ps) -> ch)   &   |- ((-. ph /\ ps) -> ch)   &   |- ((-. ph /\ -. ps) -> ch)   =>   |- ch
Theorem	niabn 764	Miscellaneous inference relating falsehoods.
|- ph   =>   |- (-. ps -> ((ch /\ ps) <-> -. ph))
Theorem	dedlem0a 765	Lemma for an alternate version of weak deduction theorem.
Theorem	dedlem0b 766	Lemma for an alternate version of weak deduction theorem.
Theorem	dedlema 767	Lemma for weak deduction theorem.
Theorem	dedlemb 768	Lemma for weak deduction theorem.
Theorem	elimh 769	Hypothesis builder for weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page.
|- ((ph <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (ch <-> ta))   &   |- ((ps <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (th <-> ta))   &   |- th   =>   |- ta
Theorem	dedt 770	The weak deduction theorem. For more information, see the Deduction Theorem link on the Metamath Proof Explorer home page.
|- ((ph <-> ((ph /\ ch) \/ (ps /\ -. ch))) -> (th <-> ta))   &   |- ta   =>   |- (ch -> th)
Theorem	con3th 771	Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This version of con3 94 demonstrates the use of the weak deduction theorem to derive it from con3i 98.
|- ((ph -> ps) -> (-. ps -> -. ph))
Theorem	consensus 772	The consensus theorem. This theorem and its dual (with \/ and /\ interchanged) are commonly used in computer logic design to eliminate redundant terms from Boolean expressions. Specifically, we prove that the term (ps /\ ch) on the left-hand side is redundant.
|- ((((ph /\ ps) \/ (-. ph /\ ch)) \/ (ps /\ ch)) <-> ((ph /\ ps) \/ (-. ph /\ ch)))
Theorem	pm4.42 773	Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Roy F. Longton, 21-Jun-2005.)
|- (ph <-> ((ph /\ ps) \/ (ph /\ -. ps)))
Theorem	ninba 774	Miscellaneous inference relating falsehoods.
|- ph   =>   |- (-. ps -> (-. ph <-> (ch /\ ps)))
Theorem	prlem1 775	A specialized lemma for set theory (to derive the Axiom of Pairing).
|- (ph -> (et <-> ch))   &   |- (ps -> -. th)   =>   |- (ph -> (ps -> (((ps /\ ch) \/ (th /\ ta)) -> et)))
Theorem	prlem2 776	A specialized lemma for set theory (to derive the Axiom of Pairing).
|- (((ph /\ ps) \/ (ch /\ th)) <-> ((ph \/ ch) /\ ((ph /\ ps) \/ (ch /\ th))))
Theorem	oplem1 777	A specialized lemma for set theory (ordered pair theorem).
|- (ph -> (ps \/ ch))   &   |- (ph -> (th \/ ta))   &   |- (ps <-> th)   &   |- (ch -> (th <-> ta))   =>   |- (ph -> ps)
Theorem	rnlem 778	Lemma used in construction of real numbers.
|- (((ph /\ ps) /\ (ch /\ th)) <-> (((ph /\ ch) /\ (ps /\ th)) /\ ((ph /\ th) /\ (ps /\ ch))))
Theorem	dn1 779	A single axiom for Boolean algebra known as DN1. See http://www-unix.mcs.anl.gov/~mccune/papers/basax/v12.pdf. (Contributed by Jeffrey Hankins, 3-Jul-2009.)
|- (-. (-. (-. (ph \/ ps) \/ ch) \/ -. (ph \/ -. (-. ch \/ -. (ch \/ th)))) <-> ch)
Abbreviated conjunction and disjunction of three wff's
Syntax	w3o 780	Extend wff definition to include 3-way disjunction ('or').
wff (ph \/ ps \/ ch)
Syntax	w3a 781	Extend wff definition to include 3-way conjunction ('and').
wff (ph /\ ps /\ ch)
Definition	df-3or 782	Define disjunction ('or') of 3 wff's. Definition *2.33 of [WhiteheadRussell] p. 105. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law orass 258.
|- ((ph \/ ps \/ ch) <-> ((ph \/ ps) \/ ch))
Definition	df-3an 783	Define conjunction ('and') of 3 wff.s. Definition *4.34 of [WhiteheadRussell] p. 118. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law anass 441.
|- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
Theorem	3orass 784	Associative law for triple disjunction.
|- ((ph \/ ps \/ ch) <-> (ph \/ (ps \/ ch)))
Theorem	3anass 785	Associative law for triple conjunction.
|- ((ph /\ ps /\ ch) <-> (ph /\ (ps /\ ch)))
Theorem	3anrot 786	Rotation law for triple conjunction.
|- ((ph /\ ps /\ ch) <-> (ps /\ ch /\ ph))
Theorem	3orrot 787	Rotation law for triple disjunction.
|- ((ph \/ ps \/ ch) <-> (ps \/ ch \/ ph))
Theorem	3ancoma 788	Commutation law for triple conjunction.
|- ((ph /\ ps /\ ch) <-> (ps /\ ph /\ ch))
Theorem	3ancomb 789	Commutation law for triple conjunction.
|- ((ph /\ ps /\ ch) <-> (ph /\ ch /\ ps))
Theorem	3anrev 790	Reversal law for triple conjunction.
|- ((ph /\ ps /\ ch) <-> (ch /\ ps /\ ph))
Theorem	3simpa 791	Simplification of triple conjunction.
|- ((ph /\ ps /\ ch) -> (ph /\ ps))
Theorem	3simpb 792	Simplification of triple conjunction.
|- ((ph /\ ps /\ ch) -> (ph /\ ch))
Theorem	3simpc 793	Simplification of triple conjunction.
|- ((ph /\ ps /\ ch) -> (ps /\ ch))
Theorem	3simp1 794	Simplification of triple conjunction.
|- ((ph /\ ps /\ ch) -> ph)
Theorem	3simp2 795	Simplification of triple conjunction.
|- ((ph /\ ps /\ ch) -> ps)
Theorem	3simp3 796	Simplification of triple conjunction.
|- ((ph /\ ps /\ ch) -> ch)
Theorem	3simp1i 797	Infer a conjunct from a triple conjunction.
|- (ph /\ ps /\ ch)   =>   |- ph
Theorem	3simp2i 798	Infer a conjunct from a triple conjunction.
|- (ph /\ ps /\ ch)   =>   |- ps
Theorem	3simp3i 799	Infer a conjunct from a triple conjunction.
|- (ph /\ ps /\ ch)   =>   |- ch
Theorem	3simp1d 800	Deduce a conjunct from a triple conjunction.
|- (ph -> (ps /\ ch /\ th))   =>   |- (ph -> ps)